- tags
- Transformers
- source
- (Xiong et al. 2021)

## Summary

This paper describes a way of applying the Nyström method for approximating matrix multiplication to transformers. More precisely, the approximation is used in the self-attention mechanism’s softmax calculation.

This approximation adresses one of the biggest downside of attention: its computational complexity. The authors claim that their method reduces it from \(O(n^2)\) to \(O(n)\).

The goal of the method is to efficiently approximate the matrix

\begin{align*} S = \text{softmax}\left(\dfrac{QK^T}{\sqrt{d_q}}\right) \end{align*}

### Why the standard Nyström method won’t work

To apply matrix approximation to this problem, one would start by writing \(S\) as

\[ S = \begin{bmatrix} A_S & B_S \\\

F_S & C_S \end{bmatrix} \]

The full matrix \(S\) can be approximated using only \(m\) columns and \(m\) rows

\[ \hat{S} = \begin{bmatrix} A_S \ F_S \end{bmatrix} A^+ \begin{bmatrix} A_S & B_S \end{bmatrix} \]